A criterion for discrete branching laws for Klein four symmetric pairs and its application to E6(−14)

Let [Formula: see text] be a noncompact connected simple Lie group, and [Formula: see text] a Klein four-symmetric pair. In this paper, we show a necessary condition for the discrete decomposability of unitarizable simple [Formula: see text]-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for [Formula: see text], there does not exist a unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. As an application, for [Formula: see text], we obtain a complete classification of Klein four symmetric pairs [Formula: see text], with [Formula: see text] noncompact, such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module and is also discretely decomposable as a [Formula: see text]-module for some nonidentity element [Formula: see text].

Connectivity and planarity of g-noncommuting graph of finite groups

Let [Formula: see text] be a finite non-abelian group and [Formula: see text] be its center. For a fixed nonidentity element [Formula: see text] of [Formula: see text], the [Formula: see text]-noncommuting graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph in which its vertices are [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we discuss about connectivity of [Formula: see text] and determine all finite non-abelian groups such that their [Formula: see text]-noncommuting graphs are 1-planar, toroidal or projective.

Generalized Torsion in Knot Groups

In a group, a nonidentity element is called a generalized torsion element if some product of its conjugates equals the identity. We show that for many classical knots one can ûnd generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the (hyperbolic) knot 52, and algebraic knots in the sense of Milnor.

Loops that are partitioned by groups

A set of subgroups of a group is said to be a partition if every nonidentity element belongs to one and only one subgroup in this set. The study of groups with partition dates back to a paper by Miller published in 1906 ([Bull. Amer. Math. Soc. 17 (1906/1907), 446–449]). In this paper we study the structure of loops that are partitioned by subgroups.

Distinguishing Maps II: General Case

A group $A$ acting faithfully on a set $X$ has distinguishing number $k$, written $D(A,X)=k$, if there is a coloring of the elements of $X$ with $k$ colors such that no nonidentity element of $A$ is color-preserving, and no such coloring with fewer than $k$ colors exists. Given a map $M$ with vertex set $V$ and automorphism group $Aut(M)$, let $D(M)=D(Aut(M),V)$. If $M$ is orientable, let $D^+(M)=D(Aut^+(M),V)$, where $Aut^+(M)$ is the group of orientation-preserving automorphisms. In a previous paper, the author showed there are four maps $M$ with $D^+(M)>2$. In this paper, a complete classification is given for the graphs underlying maps with $D(M)>2$. There are $31$ such graphs, $22$ having no vertices of valence $1$ or $2$, and all have at most $10$ vertices.

Distinguishing Maps

The distinguishing number of a group $A$ acting faithfully on a set $X$, denoted $D(A,X)$, is the least number of colors needed to color the elements of $X$ so that no nonidentity element of $A$ preserves the coloring. Given a map $M$ (an embedding of a graph in a closed surface) with vertex set $V$ and without loops or multiples edges, let $D(M)=D({\rm Aut}(M),V)$, where ${\rm Aut(M)}$ is the automorphism group of $M$; if $M$ is orientable, define $D^+(M)$ similarly, using only orientation-preserving automorphisms. It is immediate that $D(M)\leq 4$ and $D^+(M)\leq 3$. We use Russell and Sundaram's Motion Lemma to show that there are only finitely many maps $M$ with $D(M)>2$. We show that if a group $A$ of automorphisms of a graph $G$ fixes no edges, then $D(A,V)=2$, with five exceptions. That result is used to find the four maps with $D^+(M)=3$. We also consider the distinguishing chromatic number $\chi_D(M)$, where adjacent vertices get different colors. We show $\chi_D(M)\leq \chi(M)+3$ with equality in only finitely many cases, where $\chi(M)$ is the chromatic number of the graph underlying $M$. We also show that $\chi_D(M)\leq 6$ for planar maps, answering a question of Collins and Trenk. Finally, we discuss the implications for general group actions and give numerous problems for further study.

Distinguishability of Locally Finite Trees

The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$, $k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees without pendant vertices are shown to be 2-distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results are obtained for the distinguishing chromatic number, namely the least positive integer $n$ such that the function $f$ is also a proper vertex-coloring.

On The Asymptotic Values of Length Functions In Krull And Finitely Generated commutative Monoids

Let M be a commutative cancellative atomic monoid. We consider the behaviour of the asymptotic length functions and on M. If M is finitely generated and reduced, then we present an algorithm for the computation of both and where x is a nonidentity element of M. We also explore the values that the functions and can attain when M is a Krull monoid with torsion divisor class group, and extend a well-known result of Zaks and Skula by showing how these values can be used to characterize when M is half-factorial.

On doubly transitive permutation groups

Suppose that G is a doubly transitive permutation group on a finite set Ω and that for α in ω the stabilizer Gα of αhas a set σ = {B1, …, Bt} of nontrivial blocks of imprimitivity in Ω – {α}. If Gα is 3-transitive on σ it is shown that either G is a collineation group of a desarguesian projective or affine plane or no nonidentity element of Gα fixes B pointwise.

On character values in finite groups

Let u be a nonidentity element of a finite group G and let c be a complex number. Suppose that every nonprincipal irreducible character X of G satisfies either X(l) − X(u) = c or X(u) = 0. It is shown that c is an even positive integer and all such groups with c ≤ 8 are described.